A123510 - OEIS

%I #29 Sep 08 2022 08:45:28

%S 1,6,42,340,3135,32466,373156,4713192,64877805,966466270,15487707246,

%T 265617899196,4853435351947,94114052406570,1930026941433480,

%U 41728495237790416,948549349736725401,22613209058160908982,564104540143144909810,14694713818659640322340

%N Arises in the normal ordering of functions of a*(a+)*a, where a and a+ are the boson annihilation and creation operators, respectively.

%H G. C. Greubel, <a href="/A123510/b123510.txt">Table of n, a(n) for n = 0..440</a>

%F E.g.f.: (1/(1-x)^3)*exp(x/(1-x))*LaguerreL(2,-x/(1-x)), where LaguerreL(p,y) are the Laguerre polynomials.

%F From _Vaclav Kotesovec_, Nov 13 2017: (Start)

%F Recurrence: n*a(n) = 2*n*(n+2)*a(n-1) - (n-1)*(n+1)*(n+2)*a(n-2).

%F a(n) ~ exp(2*sqrt(n) - n - 1/2) * n^(n + 9/4) / 2^(3/2) * (1 + 31/(48*sqrt(n))).

%F (End)

%t max = 16; s = (1/(1-x)^3)*Exp[x/(1-x)]*LaguerreL[2, -x/(1-x)] + O[x]^(max+1); CoefficientList[s, x]*Range[0, max]! (* _Jean-François Alcover_, May 23 2016 *)

%o (PARI) m=30; v=concat([6,42], vector(m-2)); for(n=3, m, v[n]=2*(n+2)*v[n-1]-(n^2 - 1)*((n+2)/n)*v[n-2]); concat([1], v) \\ _G. C. Greubel_, May 16 2018

%o (Magma) I:=[6,42]; [1] cat [n le 2 select I[n] else 2*(n+2)*Self(n-1) - (n^2 -1)*((n+2)/n)*Self(n-2): n in [1..30]]; // _G. C. Greubel_, May 16 2018

%Y Cf. A002720, A052852, A123511, A123512.

%K nonn

%O 0,2

%A _Karol A. Penson_, Oct 02 2006

%E a(0)=1 prepended by _G. C. Greubel_, Oct 31 2017

%E More terms from _G. C. Greubel_, May 16 2018